The triple zero graph of a commutative ring

Abstract

Let R be a commutative ring with non-zero identity. We define the set of triple zero elements of R by TZ(R)=ainZ(R)ast:$thereexists$b,cinRbackslash0$suchthat$abc=0$,$abneq0$,$acneq0$,$bcneq0. In this paper, we introduce and study some properties of the triple zero graph of R which is an undirected graph TZGamma(R) with vertices TZ(R), and two vertices a and b are adjacent if and only if abneq0 and there exists a non-zero element c of R such that acneq0, bcneq0, and abc=0. We investigate some properties of the triple zero graph of a general ZPI-ring R, we prove that diam(TZGamma(R))in0,1,2 and gr(G)in3,infty.